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Let
g be a continuous function on the closed interval
[-1,3], where
g(-1)=-2 and
g(3)=-5.
Which of the following is guaranteed by the Intermediate Value Theorem?
Choose 1 answer:
(A)
g(c)=-3 for at least one
c between -1 and 3
(B)
g(c)=-3 for at least one
c between -5 and -2
(c)
g(c)=0 for at least one
c between -5 and -2
(D)
g(c)=0 for at least one
c between -1 and 3

Let $g$ be a continuous function on the closed interval $[-1,3]$ , where $g(-1)=-2$ and $g(3)=-5$ . $\newline$ Which of the following is guaranteed by the Intermediate Value Theorem? $\newline$ Choose $1$ answer: $\newline$ (A) $g(c)=-3$ for at least one $c$ between $-1$ and $3$ $\newline$ (B) $g(c)=-3$ for at least one $c$ between $-5$ and $-2$ $\newline$ (C) $g(c)=0$ for at least one $c$ between $-5$ and $-2$ $\newline$ (D) $g(c)=0$ for at least one $c$ between $-1$ and $3$

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Math Problems

Algebra 2

Domain and range of quadratic functions: equations

Full solution

Q. Let

g

be a continuous function on the closed interval

[-1,3]

, where

g(-1)=-2

and

g(3)=-5

\newline

Which of the following is guaranteed by the Intermediate Value Theorem?

\newline

Choose

1

answer:

\newline

(A)

g(c)=-3

for at least one

c

between

-1

and

3

\newline

(B)

g(c)=-3

for at least one

c

between

-5

and

-2

\newline

(C)

g(c)=0

for at least one

c

between

-5

and

-2

\newline

(D)

g(c)=0

for at least one

c

between

-1

and

3

The Intermediate Value Theorem: The Intermediate Value Theorem states that if a function $f$ is continuous on a closed interval $[a, b]$ and $N$ is any number between $f(a)$ and $f(b)$ , then there exists at least one number $c$ in the interval $(a, b)$ such that $f(c) = N$ .
Given Function and Interval: We are given that $g$ is continuous on the closed interval $[-1,3]$ , $g(-1)=-2$ , and $g(3)=-5$ . We need to find if there is a value $c$ in the interval $[-1,3]$ such that $g(c)$ equals a certain value.
Option (A) Explanation: Option (A) suggests that $g(c)=-3$ for at least one $c$ between $-1$ and $3$ . Since $-3$ is between $g(-1)=-2$ and $g(3)=-5$ , by the Intermediate Value Theorem, there must be at least one $c$ in the interval $(-1, 3)$ such that $g(c)=-3$ .
Option (B) Explanation: Option (B) is incorrect because it refers to a range of $c$ values between $-5$ and $-2$ , which are $y$ -values, not $x$ -values. The Intermediate Value Theorem applies to $x$ -values within the interval $[-1,3]$ .
Option (C) Explanation: Option (C) suggests that $g(c)=0$ for at least one $c$ between $-5$ and $-2$ , which is again referring to $y$ -values, not $x$ -values. This is not what the Intermediate Value Theorem guarantees.
Option (D) Explanation: Option (D) suggests that $g(c)=0$ for at least one $c$ between $-1$ and $3$ . However, since $0$ is not between $g(-1)=-2$ and $g(3)=-5$ , the Intermediate Value Theorem does not guarantee that there is a $c$ such that $g(c)=0$ in the interval $[-1,3]$ .